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Although popularly known as Catalan’s conjecture, this is in fact a theorem in number theory, proven by Preda Mihăilescu in 2002, 158 years after it was conjecture in 1844 by French and Belgian mathematician Eugène Charles Catalan.

Proving the Conjecture

Similarly to Fermat’s last theorem, the solution to this conjecture was assembled over a long period of time. In 1850, Victor Lebesgue established that b cannot equal 2. But then it was only after around 100 years, in 1960, that Chao Ko constructed a proof for the other quadratic case: a cannot equal 2 unless x = 3.

Hence, this left a,b odd primes. Expressing the equation as (x-1)(xa-1)/(x-1) = yb one can show that the greatest common divisor of the two left hand factors is either 1 or a. The case where the gcd = 1 was eliminated by J.W.S Cassels in 1960. Hence, only the case where gcd = a remained.

It was this “last, formidable, hurdle” that Mihailescu surmounted. In 2000, he showed that a and b would have to be a ‘Wieferich pair’, i.e. they would have to satisfy ab-1 ≡ 1 (mod b2) and ba-1 ≡ 1 (mod a2). In 2002, he showed that such solutions were impossible!

Inspired by a recent Numberphile video I thought I would share with you a cool trick to check your arithmetic is correct called ‘Casting Out Nines‘.

To check the result of a calculation using this technique, take the digital root of each number in the calculation. Then perform the calculation on these digital roots, and take the digital root of this answer. If no mistake has been made, the digital root of the result of this calculation should be the same as the digital root of the result of the original calculation.

What is a digital root?

The digital root is the value (a single digit) you get by completing an iterative process of summing the digits of the number, where you use the result from the previous iteration to compute a digit sum on each iteration.

Example: Multiplication

12345 x 67890 = 838102050

The digital roots of 12345 and 67890 are 6 and 3 respectively. The product of these digital roots is 18. The digital root of 18 is 9.

The sum of the digits of 838102050 is 27, whose digital root is also 9.

Hence, this confirms that the calculation is correct. Pretty cool eh?

How does this work?

For any number

the digit sum is . The difference between the original number and its digit sum is

Noting that numbers of the form are always divisible by 9, replacing the original number by its digit sum has the effect of casting out

Example: 3141

The Maths Behind it

Each number in the sequence uniquely determines the next number. As there are only finitely many possibilities, eventually the sequence must return to a number it has already hit. This leads to a cycle.

So any starting number will give a sequence that eventually cycles.

There can be many cycles, but for 4 digit numbers in base 10, there happens to be 1 non – trivial cycle, which involves the number 6174.

Surreal numbers were first invented by John Horton Conway in 1969, but was introduced to the public in 1974 by Donald Knuth through his book ‘Surreal Numners: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness‘.

What are Surreal Numbers?

Surreal numbers are the ‘most natural’ collection of numbers that include both real numbers and the infinite ordinal numbers of Georg Cantor. The surreals have many of the same properties as the reals, including the usual arithmetic operations. Hence, they form an ordered field.

For a surreal number x we write x = {XL|XR} and call XL and XR the left and right set of x,respectively. These will be explained below.

Conway Construction

“Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set.”

– Wikipedia

Using the Conway construction, we construct the surreal numbers in stages along with an ordering ≤ such that for any two surreal numbers a and b either a ≤ b or b ≤ a.

Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. Therefore, if x = {XL|XR} then for each xL ∈ XL and xR ∈ XR, xL is not greater than xR.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | } = 0.

The Kolakoski sequence is an infinite sequence of symbols {1,2} that is its own “run-length encoding“. It is named after mathematician Willian Kolakoski who described it in 1965, but further research shows that it was first discussed by Rufus Oldenburger in 1939.

This self-describing sequence consists of blocks of single and double 1s and 2s. Each block contains digits that are different from the digit in the preceding block.

To construct the sequence, start with 1. This means that the next block is of length 1. So we require that the next block is 2, giving the sequence 1, 2. Continuing this infinitely gives us the Kolakoski sequence: 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, etc.

The Kakeya needle problem asks whether there is a minimum area of a region in the plane in which a line segment of width 1 can be freely rotated through 360°, where translation of the segment is allowed.

This question was first posed for convex regions in 1917 by mathematician Sōichi Kakeya. It was shown by Gyula Pál that the minimum area for convex regions is achieved by an equilateral triangle of height 1 and area 1/√3.

Kakeya suggested that the minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false.

Needle rotating inside a deltoid | Source: Wikipedia

Besicovitch Sets

Besicovitch was able to show that there is no lower bound >0 for the area of a region in which a needle of unit length can be turned around. The proof of this relies on the construction of what is now known as a Besicovitch set, which is a set of measure zero in the plane which contains a unit line segment in every direction.

One can construct a set in which a unit line segment can be rotated continuously through 180 degrees from a Besicovitch set consisting of Perron trees.

Kakeya Needle Set constructed from Perron trees | Source: Wikipedia

However, although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.